The asymptotic expansion condition (REF ) can be verified using conditions in [1]}, the convergence in probability of the Hessian can be derived from a law of large numbers. The positive definiteness is a local identification condition.

Note that \(\sqrt{n}G_{nT}(\theta ^\dagger ,\Pi ^\dagger ) = \sqrt{n}[ (\ref {eq:expr1}) + (\ref {eq:expr2}) ] + \sqrt{n}(\ref {eq:expr3})\) . Assumption REF  implies that the first term converges in distribution to a Gaussian distribution with zero mean and asymptotic variance-covariance matrix \(\Sigma \) . Using the same Assumption, the second term converges in probability to zero. Using the same identity, \(\sqrt{nT} [G_{nT}(\theta ^\dagger ,\Pi ^\dagger )-(\ref {eq:expr1}) - (\ref {eq:expr2}) ] = \sqrt{nT}(\ref {eq:expr3}) \overset{p}{\rightarrow } -\kappa \mathcal {B}(\theta ^\dagger ,\Pi ^\dagger )\)  which is the first result. Then () comes from Assumption REF .
Pre-multiplying () by \(H_{nT}(\theta ^\dagger ,\Pi ^\dagger )^{-1} \overset{p}{\rightarrow } H_{\infty }(\theta ^\dagger ,\Pi ^\dagger )^{-1}\)  and using the asymptotic expansion (REF ) then implies the last result.
